#1997 ⟨a, b, c | abc=bb, cba=1⟩

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = cb
- b^-1 = d
- c^-1 = ba
- d^-1 = b
Reduced group presentation: ⟨a, b | aaaba^-1b^-1⟩
- Isomorphism: φ(a) = aaba^-1, φ(b) = a, φ(c) = b^-1

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Auxiliary generators:
1. d = ac
Reduction order:
- Left-to-right recursive path with deg(b) = deg(d) = 0, b < d; deg(a) = deg(c) = 1, a < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# abc:abc=bb,cba=1 bd/ac ac=d morph:2/1
bd=1
db=1
bba=dab
dda=bad
bc=cbbb
dc=cddd
ac=d
cba=1

5 unique, 9 total

Σ	#	Presentation	Properties	φ
8	3439	⟨a, b, c \| ba=ac, aab=c⟩	Can Inf	3
8	3465	⟨a, b, c \| ba=ac, bbb=c⟩	Can Inf	1
8	3540	⟨a, b, c \| bb=ac, cba=b⟩	Can Inf
8	5592	⟨a, b, c \| ab=c, bcca=c⟩	Can Inf
8	6110	⟨a, b, c \| ab=c, ccc=ba⟩	Can Inf

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

5 total

Σ	#	Presentation	Mapping
8	2798	⟨a, b, c \| abc=b, cbba=1⟩	φ(a) = bbad, φ(b) = b, φ(c) = c
8	4000	⟨a, b, c \| abc=1, aacba=1⟩	φ(a) = b, φ(b) = dc, φ(c) = ba
8	4035	⟨a, b, c \| abc=1, acbaa=1⟩	φ(a) = b, φ(b) = dc, φ(c) = ba
8	4098	⟨a, b, c \| abc=1, cbaaa=1⟩	φ(a) = b, φ(b) = dc, φ(c) = ba
8	4101	⟨a, b, c \| abc=1, cbbba=1⟩	φ(a) = bad, φ(b) = b, φ(c) = c